Optimal Deadlines for Agreements

ABSTRACT                                                                                                                                                               
We provide a welfare analysis of the deadline effect in a repeated negotiation game in which costly delay

can produce information that improves the quality of the decision. We characterize equilibrium strategies

and the evolution of beliefs in continuous time, and study how the length of the negotiation horizon

affects players' behavior and welfare. The optimal deadline is positive if and only if the ex ante

probability that the players disagree on the preferred decision is neither too high nor too low. When it

is positive, the optimal deadline is given by the shortest time that would allow efficient information

aggregation in equilibrium, which is increasing in the ex ante probability of disagreement and is

finitely long.

We model negotiation under a deadlines as a symmetric, continuous time repeated proposal game. 

At any instant two players simultaneously choose one of two choices to propose, paying a flow 

cost of delay, until either they agree, at which point the agreement is implemented, or the deadline 

expires and a random decision is made. The two players favor different choices: each is willing to 

go along with the other player's favorite choice only if he is sufficiently convinced that the state 

is an agreement state supporting that choice. At any point of the game, each player is either 

privately "informed," meaning that he knows the state is the agreement state corresponding to 

his favorite; or "uninformed," meaning that he is unsure whether the state is the agreement 

state corresponding to his opponent's favorite, or the state is the disagreement state with 

each player preferring his own favorite choice. 

We show that there is generically a unique equilibrium in which the informed types always 

"persist" by proposing their favorite alternative. If the deadline is sufficiently long, the behavior 

of the uninformed consists of a "concession phase," when he concedes to his opponent's favorite 

at some probability flow rate, followed by a "persistence phase," when he persists until the deadline 

is reached, at which point he plays the equilibrium strategies as in a one-shot game given his 

current belief. 

We provide a complete characterization of the "optimal deadline" that maximizes the ex ante 

probability-weighted sum of expected payoffs of the players. Naturally, the optimal deadline is 

zero when the initial belief of the uninformed about the disagreement state is sufficiently low, 

as the two players can reach the Pareto-efficient decision without delay. At the opposite end, 

with the uninformed having a sufficiently high belief about the disagreement state, the optimal 

deadline is also zero, in spite of the positive welfare effects of extending the deadline when 

it is already beyond the critical horizon. This is because a high belief of the uninformed 

means that the critical horizon is too long and the payoff loss associated with the deadline 

play too large. For intermediate initial beliefs of the uninformed, the optimal deadline is such 

that after the shortest concession phase the uninformed persists until the deadline and 

then concedes with probability one.